This by-law is quite possibly invalid, especially in the light of the “Cooper” case, which determined that blanket ban by-laws of things in a strata situation (such as pets, or split air conditioning systems I’d suggest), are not valid by-laws. Of course, there could be other factors in play, as pointed out by other respondents to your question, such as the building being designed especially to not require air conditioning, for example.
But how you deal with this (possibly invalid) by-law will depend on your situation?
I’m assuming you’re an owner and not a renter. How active and useful are your building’s strata manager, strata committee and owners? If they may be amenable to updating the by-law, then you could propose this at the next general meeting, and I’d suggest discussing this issue with the building’s strata manager, assuming there is one, and they are competent…
Otherwise, you can install the air conditioner, ignoring the by-law? This would then get the strata committee to uphold the by-law, and they might quickly discover the by-law would be invalid. If you are nervous about that, then you could get a strata lawyer to prepare an opinion about the by-law, which you could provide to the strata committee. This shouldn’t be too expensive.
Remember that the owner owns the airspace in the unit, and the balcony (depending on how old the building is, I think?), and all the Owners Corporation have a say over is common property. Which in the case of a split cycle air conditioning system is likely to be some small holes through the external walls, for the pipes and power cable) – but make sure you don’t compromise the water proofing on the balcony. Also, you need to pay attention to the water drainage, when the AC is cooling – air conditioning companies often let that water drip on to the balcony floor, which is handled by the balcony drain – assuming there is one (ie, the building is relatively modern). There might be by-laws about not letting water run off the edges of your balcony, and this is also likely to be unlawful, as creating problems for people below you. But you can collect the water produced in a bucket, or in an evaporation tray as part of the AC system. When you plan you AC system, make sure all these issues are dealt with.